Abstract

In Komatsu's work (2013), the concept of poly-Cauchy numbers is introduced as an analogue of that of poly-Bernoulli numbers. Both numbers are extensions of classical Cauchy numbers and Bernoulli numbers, respectively. There are several generalizations of poly-Cauchy numbers, including poly-Cauchy numbers with a q parameter and shifted poly-Cauchy numbers. In this paper, we give a further generalization of poly-Cauchy numbers and investigate several arithmetical properties. We also give the corresponding generalized poly-Bernoulli numbers so that both numbers have some relations.

1. Introduction

Let , be integers. Poly-Cauchy numbers of the first kind are defined by
[1]. The concept of poly-Cauchy numbers is a generalization of that of the classical Cauchy numbers defined by
(see, e.g., [2, 3]). The generating function of poly-Cauchy numbers ([1], Theorem 2) is given by
where
is the th polylogarithm factorial function. An explicit formula for ([1], Theorem 1) is given by
where are the (unsigned) Stirling numbers of the first kind, arising as coefficients of the rising factorial
(see, e.g., [4]). See ([5], A224094–A224101) for the sequences arising from poly-Cauchy numbers.

The concept of poly-Cauchy numbers is an analogue of that of poly-Bernoulli numbers [6] defined by
where
is the th polylogarithm function. When , is the classical Bernoulli number with , defined by the generating function
An explicit formula for ([6], Theorem 1) is given by
where are the Stirling numbers of the second kind, determined by
(see, e.g., [4]).

There are some kinds of generalizations of poly-Cauchy numbers. One is the poly-Cauchy number with a parameter [7] defined by
Another is the shifted poly-Cauchy number [8] defined by
Notice that can be expressed as
For example, if and , then
Therefore, such numbers are shifted from the original poly-Cauchy numbers. Remember that the Hurwitz zeta function is a generalization of the famous Riemann zeta function since .

In this paper, we give a further generalization of poly-Cauchy numbers, including both kinds of generalizations, and show several combinatorial and characteristic properties. We also give the corresponding poly-Bernoulli numbers so that both numbers have some relations.

2. Definitions and Basic Properties

Let , be integers, and let , and be nonzero real numbers. For simplicity, we write and . Define by
Then, can be expressed in terms of the Stirling numbers of the first kind .

Theorem 1. Let be a positive real number. Then,

Remark 2. If , then is the poly-Cauchy number with a parameter ([7], Theorem 1). If , then is the shifted poly-Cauchy number ([8], Theorem 2).

Proof. By
we have

For an integer and a positive real number , define the extended polylogarithm factorial function by
[8]. When , is the polylogarithm factorial function [1]. The generating function of the number () is given by using the extended polylogarithm factorial function .

Theorem 3.
One has

Remark 4. If , then Theorem 3 is reduced to Theorem 2 in [7]. If , then Theorem 3 is reduced to Theorem 3 in [8].

The generating function of the number can be written in the form of iterated integrals.

Corollary 5. Let and be real numbers with and . For , one has
For , one has

Remark 6. If , then Corollary 5 is reduced to Corollary 1 in [7]. If , then Corollary 5 is reduced to Corollary 1 in [8].

Proof. For ,
Note that the last equation holds only if is an integer. For , we have
Hence,
Putting and multiplying by , we get the result.

3. Poly-Cauchy Numbers of the Second Kind

In [1], the concept of poly-Cauchy numbers of the second kind is also introduced. The poly-Cauchy numbers of the second kind are defined by
and the generating function is given by

Then, the poly-Cauchy numbers of the second kind can also be expressed in terms of the Stirling numbers of the first kind ([1], Theorem 4). See ([5], A219247, A224102–A224107, A224109) for the sequences arising from poly-Cauchy numbers of the second kind.

Proposition 7. One has

Let be a positive real number. Similar to generalized poly-Cauchy numbers of the first kind , define the poly-Cauchy numbers of the second kind (, ) by
Then, similar to Theorem 1, can also be expressed in terms of the Stirling numbers of the first kind .

Theorem 8. One has

Theorem 9. The generating function of the number is given by
where

Remark 10. If , then Theorem 8 is reduced to Theorem 3 in [7] and Theorem 9 is reduced to Theorem 4 in [7]. If , then Theorem 8 is reduced to Theorem 5 in [8] and Theorem 9 is reduced to Theorem 6 in [8].The generating function of the number can be written in the form of iterated integrals.

Corollary 11. Let be a positive real number. For , one has
For , one has

Remark 12. When in the first identity, we have the generating function of the classical Cauchy numbers of the second kind:
In addition, there are relations between both kinds of poly-Cauchy numbers if . For simplicity, we write and .

Theorem 13. Let be an integer and a positive real number. For , one has

Proof. We will prove the second identity. The first one is proved similarly and omitted. By using the identity (see, e.g., [4], Chapter 6)
and Theorems 1 and 8, we have

4. Some Expressions of Poly-Cauchy Numbers with Negative Indices

It is known that poly-Bernoulli numbers satisfy the duality theorem for ([6], Theorem 2) because of the symmetric formula
However, the corresponding duality theorem does not hold for poly-Cauchy numbers for any real number , by the following results.

Proposition 15. Suppose that . Then, for nonnegative integers and and a real number , one has

Remark 16. If , then Proposition 15 is reduced to Proposition 1 in [7]. If , then Proposition 15 is reduced to Proposition 3 in [8].

Proof. We will prove the first identity. The second identity is proved similarly. By Theorem 3, we have

By using Proposition 15, we have explicit expressions of poly-Cauchy numbers with negative indices. For simplicity, we write and if .

Proof. By Proposition 15 together with
[4], we have
Since
we obtain
Similarly, by
we get

5. Poly-Bernoulli Numbers Corresponding to Poly-Cauchy Numbers

In this section, we will consider the corresponding generalized poly-Bernoulli numbers to the generalized poly-Cauchy numbers discussed in the previous sections. Let be an integer and a positive real number. An explicit form of poly-Bernoulli number is given by
([6], Theorem 1). In ([1], Theorem 8), one expression of in terms of poly-Cauchy numbers is given.

Proposition 19. One has

On the contrary, in ([9], Theorem 2.2), one expression of in terms of is given.

Proposition 20. One has

As a counterpart of a generalized poly-Cauchy number, we will define a generalized poly-Bernoulli number by
where is the generalized polylogarithm function defined by
so that .

Then, can be expressed explicitly in terms of the Stirling numbers of the second kind. Note that .

Proposition 21. One has

Proof. By
we have
Comparing the coefficients on both sides, we get the result.

For simplicity, we write and . If , then our results below are reduced to those previous ones.

Theorem 22. For , one has

Proof. For the first identity,
For the second identity,
Note that () and (), and

Similarly, concerning
as a generalization of poly-Cauchy numbers of the second kind , we have the following.

Theorem 23. One has

Remark 24. If , these results are reduced to the identities in Theorems 3.2 and 3.1 in [9], respectively.

Acknowledgments

This work was partly done when the first author visited Eszterházy Károly College in September 2012 and Kasetsart University in October 2012. He thanks both institute and the college for their hospitality. This work was completed when the third author visited Hirosaki University in November 2012. This work was supported in part by the Grant-in-Aid for Scientific research (C) (no. 22540005), the Japan Society for the Promotion of Science.